In previous studies on a number of under-and overdoped high temperature superconductors, including YBa 2 Cu 3 O 7−y and Tl 2 Ba 2 CuO 6+δ , the transition temperature T c has been found to change with time in a manner which depends on the sample's detailed temperature and pressure history. This relaxation behavior in T c is believed to originate from rearrangements within the oxygen sublattice. In the present high-pressure studies on HgBa 2 CuO 4+δ to 0.8 GPa we find clear evidence for weak relaxation effects in strongly underand overdoped samples (T c ≃ 40 -50 K) with an activation energy E A (1 bar) ≃ 0.8 -0.9 eV. For overdoped HgBa 2 CuO 4+δ E A increases under pressure more rapidly than previously observed for YBa 2 Cu 3 O 6.41 , yielding an activation volume of +11 ± 5 cm 3 ; the dependence of T c on pressure is markedly nonlinear, an anomalous result for high-T c superconductors in the present pressure range, giving evidence for a change in the electronic and/or structural properties near 0.4 GPa.2